Your argument at the top states
However, if someone really is the opposite gender
I do not know what you mean by this statement. I will define my argument in three stages. Genetically, physically and mentally.
Genetically speaking, No transgendered person will ever be fully male or female. No fully trans person will ever father or mother a child as the gender they transition to. Medical science has not gotten to a point where you can add or remove an X chromosome. So genetically speaking at this point in history, you will always be cross dressing.
Going away from geneticism, The physical mutilation aspect of this is key here. To get a surgery to become trans male or female, you would have to remove or add certain areas to your body, That could very easily be seen as mutilation of the body. If you don't get that surgery, You are only cross dressing which is addressed by Deuteronomy 22:5.
Mentally speaking is a trickier topic. I will confront this in the same manner as mental illness as much as it pains me to do so. Just because a person feels something does not make it true. I can feel in my heart that I am the pope. It does not make it true. A quick googling found a couple of murder cases.(as they are easier to find good examples)
Name: Johnny Frank Garrett
State: Texas
Crime: Fatally stabbed a nun in 1981
Punishment: Executed in 1992
Diagnosis: Garrett was a paranoid schizophrenic who firmly believed that lethal injection could not kill him, and that supernatural intervention by his long-dead aunt would counteract the poisons.
Name: Guy Tobias LeGrande
State: North Carolina
Crime: Killing for hire in 1993
Punishment: Sentenced to death; deemed incompetent in 2008
Diagnosis: LeGrande, who was diagnosed with a delusional disorder, killed the wife of a man who paid him to do it. He believed could communicate with Oprah through his television, and represented himself at trial wearing a Superman T-shirt. He also cursed his jury and called them "Antichrists." A state court found him incompetent to be executed. He remains in prison.
As you can see above believing something does not make it true just as equally as not believing something does not make it true. Russell's Teapot should be used more often, in that the burden of proof should lie on those who make absurd claims to prove them as opposed to making others disprove them.
I am an atheist playing devils advocate(Pun intended). None of these views are my own, and in fact I just got back from a drag show last night. I fully support the LGBT lifestyle.